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Microwave Imaging for Conducting Scatterers by Hybrid Particle Swarm Optimization with Simulated Annealing

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Microwave Imaging for Conducting Scatterers by Hybrid Particle Swarm Optimization with Simulated Annealing 2011 8th International Multi-Conference on Systems, Signals & Devices MICROWAVE IMAGING FOR CONDUCTING SCATTERERS BY HYBRID PARTICLE SWARM OPTIMIZATION WITH SIMULATED ANNEALING Bouzid Mhamdi, Khaled Grayaa and Taoufik Aguili Syscom Laboratory, Engineer School of Tunis - BP 37 Belvedere, Tunis - 1002, Tunisia [email protected] ABSTRACT The first is based on gradient searching schemes such as the Newton-Kantorovitch method [3], modified In this paper, a microwave imaging technique for gradient method [4] and Levenberg-Marguart algorithm reconstructing the shape of two-dimensional perfectly [5]. It is well-known that these deterministic techniques conducting scatterers by means of a stochastic used for fast reconstruction of microwave images tend to optimization approach is investigated. get trapped in a local extreme and suffer from a major Based on the boundary condition and the measured drawback, where the final image is highly dependent on scattered field derived by transverse magnetic the initial trial solution. To overcome this obstacle, a illuminations, a set of nonlinear integral equations is second class of population based stochastic methods such obtained and the imaging problem is reformulated in to as the genetic algorithm (GA) [1-2], simulated annealing an optimization problem. (SA) [6] and neural network (NN) [7] has become A hybrid approximation algorithm, called PSO-SA, is attractive alternatives to reconstruct microwave images. developed in this work to solve the scattering inverse These techniques consider the imaging problem as a problem. In the hybrid algorithm, particle swarm optimization (PSO) combines global search and local global optimization problem, and reconstruct the correct search for finding the optimal results assignment with image by searching for the optimal solution through the reasonable time and simulated annealing (SA) uses use of either rivalry or cooperation strategies in the certain probability to avoid being trapped in a local problem space. optimum. Recently, the particle swarm optimization (PSO) The hybrid approach elegantly combines the exploration algorithm [8] has attracted the interest of the ability of PSO with the exploitation ability of SA. electromagnetic community as a promising technique for Reconstruction results are compared with exact shapes of the solution of optimization and design electromagnetic some conducting cylinders; and good agreements with the problems [9]. Actually, the PSO algorithm has already original shapes are observed. been applied to the solution of inverse scattering problems related to the reconstruction of dielectric Index Terms— Hybrid algorithms, Microwave scatterers [10-11]. imaging, Particle Swarm Optimization, Simulated It has been reported that PSO has better performance Annealing, Shape reconstruction in solving some optimization problems. However, basic PSO algorithm suffers a serious problem that all particles are prone to be trapped into the local minimum in the later phase of convergence. Although the SA Algorithm 1. INTRODUCTION has been proved that it could convergent to overall optimal solution by using probability, but more time is The application of electromagnetic scattering to retrieve taken in seeking high-quality approximately optimal the shape, location, size, and the internal property of an solution [12]. object embedded in a homogeneous space or buried In this paper, we proposed a new hybrid optimization underground has gained increasing interest in many areas approach which combines PSO algorithm with the SA such as non-destructive evaluation, geophysics, algorithm and apply it to solve the scattering inverse biomedical applications, material engineering, remote problem. By integrating SA to the PSO, the new sensing, and environmental investigations. A particular algorithm, which we call it PSO-SA makes full use of the problem of this kind is the estimation of the location as strong global search ability of PSO and the strong local well as the shape of conducting scatterers [1-2]. search ability of SA and offsets the weaknesses of each However, it is well known that the main difficulties other. of inverse problems are the nonlinearity and ill-posed. As The PSO-SA approach is applied to the a result, many inverse problems are reformulated as reconstruction of the shape of perfectly conducting optimization problems. Much attention has been paid to scatterers. The principal objective of the proposed developing the inversion algorithms and a variety of technique is to minimize a cost function that describes the algorithms have been proposed. Generally speaking, two discrepancy between the measured and estimated main kinds of approaches have been developed. scattered field data. 978-1-4577-0411-6/11/$26.00 ©2011 IEEE Measurement Incident 2. PROBLEM FORMULATION 2.1. Forward problem The geometrical configuration is depicted in figure 1. We consider a Perfectly Electrically Conducting (PEC) cylinder with circular cross section placed in a homogeneous background medium. The cylinder is assumed infinite long in z direction, while the cross-section of the cylinder is arbitrary, so a 2D approximation can be used. The scatterer is illuminated by a circumference source of pulsed plane waves, with z-polarized incident electric field El (TM polarization) and a propagation direction impinges at an incident angle p l in the (x, y) plane. The z polarized incident wave can be given by (1): Figure 1. Geometry of the considered inverse scattering El(x,y) — g-jKo&cosvi+y.sinvi)' ( 1) problem in (x, y) plane. Where where Ko — w ju o- eo denotes the free-space wave number. 8 — / p 2(p) +~p2(pTf—'2pXpJpXp') cos(p—~pT) (8) In this work, the shape of the object can be expressed by means of a Fourier series of order Q as follow: f — J p 2(y') + p'2(p") (9) Conventionally, (7) is solved by employing the p (p ) — 1 an cos(np) + 1 bn sin(n^) (2) method of moments (MoM) [13] in which the contour of the cylinder is divided into R segments. The current n = 0 n = l density is assumed to be constant in each segment and a an and bn represent the Fourier coefficients: matrix equation is derived to solve for the unknown current coefficients. One should not forget that the direct 1 f 2n scattering problem is to calculate the scattered electric an — — I p(p) cos(np) dp (3) fields while the shape and location of the scatterer are n -*0 given. Then, the scattered field is given as: 1 f 2n IX bn — I p (p ) sin(n^) dp (4) MUo V "1 n Jo Es(r,p) — ---- 4 ~ A p 1 jn H (<0l)(Ko8n)fn (10) The total and scattered electric fields are polarized along the z-axis, thus the problem is reduced to a TM Where Ap is the angular discretization size, 8n and fn are scalar formulation. Starting from Maxwell’s equations, a given as follow: Freedom integral equation of first kind may be derived in which the scattered field EsSz is given as: 8n — jr 2 + p2(pn) — 2rp(pn) COS(p — pn) (11) Es(r) — - ^ 0 j Jz(r')H0(Ko\r — f'|)df' (5) fn — j p2(pn) + p'2(pn) (12) 2.2. Scattering inverse problem Where r and r denote the observation and field points, Jz is the induced current density parallel to the z-axis, and Let us assume that the scatterer is illuminated by a H1 is the Hankel function of the first kind and zeroth number of incident fields and that for each incidence the order. At the surface of the cylinder, the total electric scattered field is measured at a set of measurement field satisfies the boundary condition: positions around the scatterer domain. The objective of the microwave imaging, i.e. the inverse problem is to E l + E S — 0 (6) estimate the scatterer contour from the total set of measurements. To cope with this inverse scattering Considering that r — (p(p),p) on counter C, if we problem, we define a cost function representing the combine (5) and (6), one obtains: discrepancy between measured and estimated scattered field. In this study, the inversion procedure is based on MUo f , „ El(p(p'),p') —4° | Jz(p)Ht(Ko8)fdp (7) minimization of the cost function F which represents a relative error with respect to the Fourier coefficients. In each iteration, the velocity and position of each S meas s 2 1 V V \^im ^im\ F(X)_ (13) particle are updated according to its best encountered meas \ position and the best position encountered by any \E<i \ particle, in the following way: X _ [a0, ai,..., aQ, bv b2 ,...,bQ] Vi.d _ w. Vu + Ciri(Pi,d - Xiid) + C2rz{Pg,d - Xi,d)(14) Where S is the total number of incidences, M is the Xi,d _ Xi,d + Vi,d (15) number of measurements per incidence, Emeas denotes the measured scattered field and Es is the estimated one. where w is the inertia weight. cl and c2 are the Actually, since the contour is a function of X, (13) is acceleration coefficients and the parameters rl and r2 are minimized with respect to X. two random numbers distributed uniformly in [0,1]. In this work, this problem is resolved by an Generally, the value of Vid is restricted in the interval optimization approach, for which the global searching [-Vmax,Vmax], Vmax is decided by the user. scheme PSO and its hybrid model with the SA heuristic The inertia w is used to achieve a balance in the method are employed to minimize the above cost exploration and exploitation of the search space. The function (F). inertia dynamically reduces during a run which facilitates a balance in the exploration and exploitation of the search 3. RELATED ALGORITHMS space. 3.1. Particle Swarm Optimization 3.2. Simulated annealing Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Eberhart Simulated annealing (SA) was proposed by S.
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